To state our main results, we first recall Colding–Minicozzi's definition [1]} of entropy of \(M^n\subset \mathbb {R}^{n+1}\) :
\(\lambda (M) := \sup _{\begin{array}{c}\mathbf {x}_0\in \mathbb {R}^{n+1}\\t_0>0\end{array}}\int _{M} (4\pi t_0)^{-\frac{n}{2}} e^{-\frac{1}{4t_0} |\mathbf {x}- \mathbf {x}_0|^{2}} .\) 
